Degree-three irrationality conjecture for eight Fano threefold deformation families

Let $X_d$ be a quasismooth member of one of the eight deformation families numbered \textnumero 1, \textnumero 19, \textnumero 28, \textnumero 39, \textnumero 49, \textnumero 59, \textnumero 66, and \textnumero 84. Let $\mathrm{d}(X_d)$ denote the smallest degree of a generically finite dominant rational map from $X_d$ to projective space of the same dimension. Degree-three irrationality conjecture. For every such $X_d$,

$$\mathrm{d}(X_d)=3.$$

The surrounding theorem states this for a general member of each of the eight families, while the candidate extends the assertion to every quasismooth member; the status of this stronger formulation is not established by the supplied context.