It's probably useful to take a few steps back first, since this text is from Chandrakirti, an author which views differ significantly from some of the core principles of other schools in his branch of Buddhism.
The linked text originates in the Madhyamaka school of Nagarjuna, which means that as a basic principle, author will view all phenomena as ultimately empty of intrinsic nature or self-existence (svabhāva). Furthermore, all things (bhāva) have two natures, the conventional and the ultimate
As such, the conventional truth (saṁvṛti satya) implies that, as a matter of human convention, phenomena have a nature or existence (bhāva), for example ice has the property of being cold. The world-view of conventional truth is the building block of human communication that allows us to build societies.
As for ultimate truth, well, this is where Madhyamaka branches off in different schools of thought, like Sautrāntika, Kamalaśīla, Yogācāra and Prāsaṅgika (Chandrakirti belongs to the latter). It would take way too long to detail the differences here. If you're interested, The Theory of Two Truths in India is an excellent read.
In essence, Candrakīrti rejects the theories of the two truths in both Brahmanical and the Buddhist schools of Vaibhāṣika, Sautrāntika, Yogācāra and Svātantrika Madhyamaka. At the core of the Prāsaṅgika's theory of the two truths are these two fundamental theses:
Only what is conventionally non-intrinsic is causally effective, for only those phenomena, the conventional nature of which is non-intrinsically real, are subject to conditioned or dependent arising. Conventional reality (or dependently arisen phenomenon), given it is causally effective, is therefore always intrinsically unreal. Hence that which is conventionally (or dependently) coarisen is always conventionally (or dependently) arisen.
Only what is ultimately non-intrinsic is causally effective, for only those phenomena, the ultimate nature of which is non-intrinsically real, are subject to conditioned arising. Ultimate truth (or emptiness), given it is causally effective, is therefore intrinsically unreal. Hence ultimate truth is ultimately unreal (or emptiness is always empty)
Madhyamaka schools generally state that all phenomena are merely conceptual constructs (prajñaptimatra) which do not exist in themselves but are mentally constructed dependent designations (prajñāptirupādāya). However, Chandraktiri views even conventional truth (which is not a phenomenon, but a mental state or construct) as empty of intrinsic natures. As such, he defines ultimate truth as:
"the nature of things found by particular exalted cognitive processes
(yeshes) of those who perceive reality."
Now with this in consideration, let's take a look at your statement:
The conventional self is not different to all its parts (is composed
only of parts) but not the same as any part (does not depend on any
part)...this arguably, is a reason to think that the self cannot be destroyed
Candrakīrti attributes three senses to the term convention:
confusion (avidyā) for it obstructs the mundane cognitive processes of the ordinary beings from seeing the reality as it is by way of concealing (saṃvṛti) its nature;
codependent (paraparasaṃbhavana) for it is an interdependent phenomenon
Signifier (saṁket) or worldly convention (lokavyavahāra) in the sense of dependently designated by means of expression and expressed, consciousness and object of consciousness
All 3 senses can be applied to your statement, giving very different renditions pointing to the same underlying conventional truth. For example:
the conventional self, like all phenomena, is subject to constant change. Its acclaimed parts are therefore also subject to change. Since there is no "constant" conventional self, there is no "constant" that cannot be destroyed.
if the conventional self arises dependent on its parts, then essentially there is no intrinsic self that cannot be destroyed.
as a worldly convention, we interact with our and other "selves" as mental constructs for convenience of communication. Nothing in this convention however points to these "selves" being indestructible.
Remember that all these constructs are built upon conventional reality, which Chandraktiri views as empty of intrinsic natures.
