Infinitude conjecture for primary Carmichael numbers

Let $s_p(m)$ denote the sum of the base-$p$ digits of $m$, and let $4\mathcal{C}'4$ be the set of squarefree numbers $m$ such that $s_p(m)=p$ for every prime factor $p$ of $m$. Let $4\mathcal{C}4$ be the set of Carmichael numbers.

Infinitude conjecture for primary Carmichael numbers. The following claims are true: the set $4\mathcal{C}'4$ is infinite, and the set $4\mathcal{C}\setminus\mathcal{C}'4$ is infinite.

The conjecture is supported by numerical data showing a slow but steady increase in the counting function for $4\mathcal{C}'4$. It would imply another proof of the infinitude of Carmichael numbers and of the infinitude of the relevant squarefree set.