Irreducibility conjecture for symmetric-group summands of torus-link homology

Let $T_n=T(n,n)\subset S^3$ and let $v_{C,n}$ be the lattice point where the $C^{th}$ and $(C+1)^{th}$ Alexander-support hypercubes meet. Assume the Heegaard Floer link homology is taken over a field of characteristic zero, so the symmetric-group action conjecture supplies an $S_n$-representation on the homology. Irreducibility conjecture. Each graded summand of

$$\widehat{HFL}(S^3,T_n,v_{C,n})$$

is an irreducible representation of $S_n$. This proposes a representation-theoretic refinement of the extremal-vertex homology conjecture: its binomial-dimensional graded summands match the dimensions of hook representations in characteristic zero. The assertion remains open, and the characteristic-two case is explicitly more subtle.