Weight preservation for deletion-confusable Naisargik images of quaternary VT codes

Let $VT_{(a,b)}(n;4)$ be a quaternary Varshamov–Tenengolts code, let $\phi$ be a Naisargik map, and let $D_1(X)$ denote the set of binary sequences obtained from $X$ by one deletion. For a binary sequence $X$, write $w_X$ for its weight. Weight-preservation conjecture. For all $X,Y\in\phi(VT_{(a,b)}(n;4))$, if

$$D_1(X)\cap D_1(Y)\ne\emptyset,$$

then $w_X=w_Y$. The Naisargik map $\phi$ denotes all maps from $\phi_1$ to $\phi_8$ specified in the paper. The claim concerns whether the Naisargik image has the deletion error-correction property; the authors report exhaustive computational evidence and a partial proof, but no comprehensive proof is currently available.