Tanisaki quotient conjecture for the modules $V_n^=({\mathbf a})$

Let $r\leq n$ be nonnegative integers, put $k=n-r$, and let ${\mathbf a}\in({\mathbb Z}_{\geq0})^r$. Define $V_n^=({\mathbf a})$ as the associated graded quotient for the componentwise partial order on length-$r$ sequences, and let $R_\lambda$ be the Tanisaki quotient associated with a partition $\lambda$. Let ${\mathrm {grFrob}}$ denote graded Frobenius image, $\omega$ the standard involution on symmetric functions, and ${\mathrm {rev}}_q$ reversal in $q$. Tanisaki quotient conjecture. There exists a partition $\lambda\vdash n$ with $k$ parts such that

$$({\mathrm {rev}}_q\circ\omega){\mathrm {grFrob}}(V_n^=({\mathbf a});q)={\mathrm {grFrob}}(R_\lambda;q).$$

Equivalently, if $Q'_\lambda(X;q)$ is the Hall–Littlewood $Q'$-function, then

$$\omega[{\mathrm {grFrob}}(V_n^=({\mathbf a});q)]\propto Q'_\lambda(X;q),$$

where $\propto$ denotes equality up to a power of $q$. The claim would extend the observed relationships between these modules and Tanisaki quotients beyond the computed examples; the source explicitly notes that it does not have a full conjecture in this direction, so this formulation is computationally motivated.