Infinitude conjecture for the sets of digit-sum classes

For each integer $d\geq2$, let $\mathcal{S}_d$ be the set defined in the source, and let $\mathcal{S}$ be their covering set. Let $\mathcal{C}$ denote the set of Carmichael numbers.

Infinitude conjecture for the sets of digit-sum classes. For each $d\geq2$, the sets $\mathcal{S}_d$ and $\mathcal{S}\setminus\mathcal{S}_d$ are infinite. Moreover, $\mathcal{S}\setminus\mathcal{C}$ is infinite.

The conjecture concerns the distribution of the classes $\mathcal{S}_d$ within the covering set $\mathcal{S}$ and predicts that $\mathcal{S}$ contains infinitely many non-Carmichael numbers. The source does not provide a resolution.