The continuous multi-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 $\mathbf{k}=(k_1\ge\cdots\ge k_r\ge 1)$ satisfy

$$n>k_1>k_2+1>k_3+2>\dots>k_r+r-1>r-1.$$

For $0\le q_1<\dots<q_n<1$, the continuous multi-away density conjecture.

$$g_{s_{k_1}\dots s_{k_r}w_0}=\left(\frac{1}{k_r!}\frac{\partial^{k_r}}{\partial q_{n-k_r+1}\dots\partial q_n}-1\right)g_{s_{k_1}\dots s_{k_{r-1}}w_0}.$$

This conjecture translates the multi-away multiline-queue enumeration formula into a recursive relation for continuous stationary densities. The paper reports no proof of this general relation.