Infinitude conjecture for elliptic curves with odd modular degree

An elliptic curve over $\mathbf{Q}$ has odd modular degree when the degree of its optimal modular parametrization is odd. The odd modular-degree infinitude conjecture. There are infinitely many elliptic curves over $\mathbf{Q}$ with odd modular degree. The paper motivates this with the conjectural infinitude of primes of the form $(8n+3)^2+64$ and presents it as an open conjecture supported by data.