Inseparable endomorphisms and rank-2 sublattices of the Gross lattice

We answer a question posed by Love asking about a correspondence between isogenies from a supersingular elliptic curve to its Frobenius base-change and rank-2 sublattices of its Gross lattice. We recast the question as one about the inseparable endomorphisms of the curve, and show that the correspondence holds when the trace of the endomorphism is zero, and may not hold otherwise.