A variation on local trilinear form

Let $F$ be a $p$-adic field and $M$ the subgroup of ${\mathrm{G}\mathrm{L}}_2^3(F)$ consisting of triples with equal determinants. For each unitary principal series $L$-packet of $M$ whose central character is trivial on the diagonal $F^{\times }$, we determine its unique member distinguished by the diagonal ${\mathrm{G}\mathrm{L}}_2(F)$.