Elaborating the word problem for free idempotent-generated semigroups over the full transformation monoid

With each semigroup one can associate a partial algebra, called the biordered set, which captures important algebraic and geometric features of the structure of idempotents of that semigroup. For a biordered set $\mathcal{E}$, one can construct the free idempotent-generated semigroup over $\mathcal{E}$, $\mathsf{IG}(\mathcal{E})$, which is the free-est semigroup (in a definite categorical sense) whose biorder of idempotents is isomorphic to $\mathcal{E}$. Studies of these intriguing objects have been recently focusing on their particular aspects, such as maximal subgroups, the word problem, etc. In 2012, Gray and Ru\v{s}kuc pointed out that a more detailed investigation into the structure of the free idempotent-generated semigroup over the biorder of $\mathcal{T}_n$, the full transformation monoid over an $n$-element set, might be worth pursuing. In 2019, together with Gould and Yang, the present author showed that the word problem for $\mathsf{IG}(\mathcal{E}_{\mathcal{T}_n})$ is algorithmically soluble. In a recent work by the author, it was showed that, for a wide class of biorders $\mathcal{E}$, the algorithmic solution of the word problem revolves around the so-called vertex groups, which arise as certain subgroups of direct products of pairs of maximal subgroups of $\mathsf{IG}(\mathcal{E})$. In this paper we determine these vertex groups for the case when $\mathcal{E}$ is the biorder of idempotents of $\mathcal{T}_n$.