Uniqueness of the degree greater than one morphism from $X_0(p)$

Let $p$ be a prime. Write $X_0(p)^+$ for the quotient of the modular curve $X_0(p)$ by its usual involution. Uniqueness conjecture. Up to isomorphism, the quotient map

$$X_0(p) \to X_0(p)^+$$

is the only non-constant rational morphism of degree $d>1$ from $X_0(p)$ to a curve of genus at least $2$. This conjecture is motivated by the stated results for primes below $3000$ and for primes whose Jacobians satisfy the given decomposition hypothesis; the numerical evidence in the source suggests that the decomposition hypothesis applies to an increasing proportion of prime levels, but the assertion for all primes remains open.