On the use of subsequential symbol $ in Finite state transducers to pad out the context, for composition

Question

I am going through hbka.pdf (WFST paper - https://cs.nyu.edu/~mohri/pub/hbka.pdf)

I am trying to understand this paragraph. Pg 22/31.

"The deterministic transducer composes without a matching delay, which makes it the better choice in applications. However, it introduces a single-phone shift between a context-independent phone and its corresponding context-dependent unit in the result. This shift requires the introduction of a final subsequential symbol $ to pad out the context. In practice, this might be mapped to a silence phone or an epsilon-transition."

I understand that "matching delay" here does not mean time delay or any mismatch. It refers to the "computational steps" involved in finding the correct alignment. I also see that single-phone shift, for example is the mismatch between 'a' and 'a', but both a's have happened in different 'phonetic' contexts; like /aa/ and /ae/.

Figure for reference

I do not see how exactly useful is this $-subsequential symbol in padding out the context. Why pad out (remove) the contexts at all; they provide additional information.