The upper-bound conjecture for reduced Weierstrass curves over prime-power rings

Let $p$ be an odd prime and let $m$ be a positive integer with $\gcd(p^m,6)=1$. Let $\Delta^i(p^m)$ denote the number of solutions to $\Delta \equiv i \pmod {p^m}$, and let $N_R(\mathbb{Z}_{p^m})$ be the number of nonsingular reduced Weierstrass elliptic curves over $\mathbb{Z}_{p^m}$. Reduced Weierstrass upper-bound conjecture. The upper bound $N”_R(\mathbb{Z}_{p^m})$ is

$$N”_R(\mathbb{Z}_{p^m})=p^{2m}-\Delta^0(p^m)\geq N_R(\mathbb{Z}_{p^m}).$$

The claim is presented as a computationally supported bound; no resolution status is supplied.