Threshold-shuffle identities for uri multiplicities

Prove that the uri multiplicities μ_{a,α}=B_1(ℓ(α),ind_a(α)) satisfy the threshold-shuffle identities defined for the composition multiplicities, thereby establishing the uri post-Lie structure unconditionally.

Background

The uri triangle map is defined using multiplicities derived from Bernoulli numbers. Theorem 4.?? proves that these multiplicities produce a post-Lie algebra only under the threshold-shuffle identities. The conjecture is therefore the principal unresolved combinatorial condition needed to remove that assumption and obtain the uri post-Lie and associated Hopf-algebra structures unconditionally.