Elliptic curves of conductor $2^{m}p$, quadratic twists, and Watkins' conjecture

Let $\mathsf{E}\mathrm{/}\mathbb{Q}$ be an elliptic curve. By the modularity theorem, it admits a surjection from a modular curve $X_0(N)\to \mathsf{E}$, and the minimal degree among such maps is called the modular degree of $\mathsf{E}$. By the Mordell–Weil Theorem, $\mathsf{E}(\mathbb{Q})\simeq {\mathbb{Z}}^r\oplus T$ for some nonnegative integer $r$ and some finite group $T$. Watkins’ Conjecture predicts that $2^r$ divides the modular degree, thus suggesting an intriguing link between these geometrically- and algebraically-defined invariants. We offer some new cases of Watkins’ Conjecture, specifically for elliptic curves with additive reduction at $2$, good reduction outside of at most two odd primes, and a rational point of order two.