Uniqueness conjecture for Markov numbers of rigid strings

Let $C=(x_1:a_1,a_2,\ldots,a_n)$ and $C'=(x'_1:a'_1,a'_2,\ldots,a'_m)$ be two $E_{\Lambda}$-rigid strings. For a string $D$, let $m(D)$ denote the sum of Euler characteristics of the Grassmannians of subrepresentations of its associated string module.

Markov-number uniqueness conjecture. One has

$$m(C)=m(C')$$

if and only if $n=m$ and $a_i=a'_i$ for all $1\leq i\leq n$.

This conjecture asserts that the Markov number associated with an $E_{\Lambda}$-rigid string determines the length and sequence of its string data, independently of the initial vertex. The source gives no evidence of a resolution.