The supersingular-prime congruence for elliptic-curve poles

Let $C/\mathbb{Q}$ be an elliptic curve with $j(C)\notin\{0,1728\}$. A prime $p$ is supersingular for $C$ when $a_p(C)=0$ (for $p\geq5$), where $a_p(C)=p+1-|C(\mathbb{F}_p)|$. Let $v_p$ denote the $p$-adic valuation. The supersingular-prime congruence. If $p$ is a supersingular prime of $C$ and $v_p(j(C))=0=v_p(j(C)-1728)$, then

$$a_p\left(\frac{E_4}{j-j(C)}\right)\equiv0\pmod{p^2}.$$

The preceding theorem gives only the congruence modulo $p$; this stronger modulo-$p^2$ assertion is presented as a numerical observation and remains open.