If the term 'center' in the question refers to the Middle Way then I'd say the answer is no.
Relative truths are never ultimate truths so immediately there's a problem with the question. Ultimate truths seem paradoxical, as Lao Tsu notes, while relative truths are not truths in a metaphysical sense.
Take the famous comment by Heraclitus 'We exist and exist-not'. This would be a rigorous statement according to Nagarjuna but not 'true'. There would be no such thing as a true ultimate statement since Reality is beyond the reach of linguistic/conceptual fabrications. Dialectical logic cannot decide the truth-values of paradoxical statements.
The reason remarks like this one of Heraclitus are not true is that for an ultimate view there is no Middle Way. A 'Middle Way' assumes two extreme views between which is the Way, but in the very final analysis Nagarjuna rejects the extremes therefore also the idea there is a 'middle way' between them. Thus Heraclitus can combine two relative statements, each of which is true in a sense, for an ultimate statement that is rigorous. He makes no mention of a middle way, however, but juxtaposes two half-truths to show the inadequacy of both.
This is the via negativa, the avoidance of making positive statements about the world, necessary for the Middle Way view because all positive or extreme metaphysical theories would be wrong.
Briefly - Relative truths are never ultimately true. Ultimate truths cannot be stated but may be indicated in rigorous language. No statement is both relatively and ultimately true. All relative truths are not ultimate truths. Truth outruns language (because of its subject/predicate structure) and conceptualisation (because it requires the categories of thought, which are not fundamental).
Thus all relative truths would be untrue for an ultimate analysis and all ultimate truths would be unsayable, but may be 'pointed at' by the use of a language of contradiction that combines opposing relative truths.
The need to combine two relative truths for a rigorous ultimate statement means that a relative truth is never an absolute truth. Statements that are in the 'center' may be rigorous but are never simply true or false. The notion of 'true' and 'false' belong to the realm of relative statements. An ultimate statement will seem paradoxical, not simply true or false.
So all in all the question is formulated in a way that does not allow a simple answer. This one is hasty and may not be at all clear but perhaps it indicates the various relevant issues that need to be considered.
I'm not sure if it's relevant to say this, but I think that "Middle" Way is classically used to deny (or to be in contrast to, different than) the two extremes:
Mendicants, these two extremes should not be cultivated by one who has gone forth. What two? Indulgence in sensual pleasures, which is low, crude, ordinary, ignoble, and pointless. And indulgence in self-mortification, which is painful, ignoble, and pointless. Avoiding these two extremes, the Realized One woke up by understanding the middle way, which gives vision and knowledge, and leads to peace, direct knowledge, awakening, and extinguishment.
Similarly it's neither eternalism nor annihilationism.
Or neither lust, nor hatred.
The PTS dictionary says,
Applied almost exclusively in contrast pairs with terms of more or less, in triplets like "small -- medium -- big," or "first -- middle -- last"
I suppose that "middle" in this sense doesn't mean "half-way in between" -- it means "neither one nor the other".
Yes, always. Ultimate has only ultimate as relative. While relative has only relative as relative. Understand? There is no more relation to relative once ultimate has reached, Nyom.
See also Change-of-lineage for more.
The tradition is called "tradition of the Noble Ones (ariya puggala)".
(Note that this is not given for exchange, stacks, trade end entertain relations but for escape from bounds toward world.)
The Buddhist Roots of Zhu Xi's Philosophical Thought
I'm not sure, but would guess that yes in the exclusive middle, but not in the non-exclusive middle.
Often, different interpretations of the 'middle' demarcate the different teachings. So perhaps something relatively and ultimately true can be false in the center, just not for its perfect expression.