The continuous one-away TASEP density conjecture

Let $g_w(q_1,\dots,q_n)$ denote the stationary probability density for the continuous TASEP configuration with permutation $w$, and let $w_0$ be the longest permutation. For $n>k\ge 1$ and $0\le q_1<\dots<q_n<1$, the continuous one-away density conjecture.

$$g_{s_kw_0}=\left(\frac{1}{k!}\frac{\partial^k}{\partial q_{n-k+1}\dots\partial q_n}-1\right)g_{w_0}.$$

This is the continuous-distribution counterpart of the one-away multiline-queue enumeration conjecture, obtained by passing to the scaling limit. The source presents it as a conjectural consequence rather than a proved general formula.