Shift-invariance conjecture for the KPZ equation

Fix $t>0$, $\Delta>0$, two segments $[X,X']$ and $[Y,Y']$ in $\mathbb R$, and real numbers $(x_1,\dots,x_n)$, $(y_1,\dots,y_n)$, $(\tilde x_1,\dots,\tilde x_k)$, and $(\tilde y_1,\dots,\tilde y_k)$. Assume that for each $1\le j\le k$, $[\tilde x_j,\tilde x_j+\Delta]\subset[X,X']$ and $[\tilde y_j,\tilde y_j+\Delta]\subset[Y,Y']$. Also assume that for each $1\le i\le n$, either $x_i\le X$ and $y_i\ge Y'$, or $x_i\ge X'$ and $y_i\le Y$. Let $\mathcal Z^{(y)}(t,x)$ denote the KPZ equation's continuum directed random polymer partition function from $y$ to $x$ at time $t$. KPZ shift-invariance conjecture. The joint distribution satisfies

$$\bigl(\mathcal Z^{(\tilde y_j)}(t,\tilde x_j), j=1,\dots,k; \, \mathcal Z^{(y_i)}(t,x_i), i=1,\dots,n \bigr)\stackrel{d}{=}\bigl(\mathcal Z^{(\tilde y_j+\Delta)}(t,\tilde x_j+\Delta), j=1,\dots,k; \, \mathcal Z^{(y_i)}(t,x_i), i=1,\dots,n\bigr).$$

The conjecture extends the known shift-invariance theorem for configurations satisfying $\tilde x_1<\cdots<\tilde x_k$ and $\tilde y_1>\cdots>\tilde y_k$; the claimed $k$-dimensional invariance without these inequalities is motivated by numerical and formulaic evidence and remains unresolved here.