The Delta Conjecture skewing formula for the quadratic t coefficient

Let $n$ and $k$ be the parameters in the Delta Conjecture, let $s_\lambda^\perp$ denote the operator adjoint to multiplication by $s_\lambda$, let $H_\lambda(x;q)$ be the Hall–Littlewood polynomial, and let $[r]_q$ and $\binom{r}{m}_q$ denote the usual $q$-analogues. Quadratic-coefficient skewing conjecture. The $t^2$ coefficient of $\omega\Delta'_{e_{k-1}}e_n$ is

$$s_{((n-k)^{k-1})}^\perp\bigg([k-2]_qH_{(n-k+2,(n-k+1)^{k-2},n-k)}(x;q)+\binom{k-2}{2}_qH_{((n-k+2)^2,(n-k+1)^{k-4},(n-k)^2)}(x;q)+[k-1]_qH_{(n-k+3,(n-k+1)^{k-2},n-k-1)}(x;q)\bigg).$$

This is part of the paper’s proposed skewing formulas for low powers of $t$, extending the known $t=0$ formula and yielding positive Schur expansions after expanding Hall–Littlewood polynomials. Its resolution is not specified in the source.