The weighted congruence deduction conjecture

Let $\bs=(s_1,\ldots,s_k)$ and $\bt=(t_1,\ldots,t_m)$ be compositions, and identify compositions with words in non-commuting symbols $x,y$ that do not end in $x$. Let $h_p(\bs\s\bt)$ denote the sum of $h_p(\bs_3)$ over the shuffles of the words corresponding to $\bs$ and $\bt$. Jarossay's double-shuffle identity gives a convergent $p$-adic series relation for these shuffled weighted multiple harmonic sums.

Weighted congruence deduction conjecture. Every weighted congruence can be deduced from Theorem.

The claim asserts that the stated family of double-shuffle identities is sufficient to generate all weighted congruences. The supplied text does not state whether this sufficiency has been established beyond the indicated deduction procedure.