A variation on local trilinear form

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