Congruence progression conjecture for primary pseudoperfect numbers

Let $K_r$ denote a primary pseudoperfect number with $r$ prime factors. The claim concerns those $K_r$ divisible by $6$. Primary pseudoperfect congruence conjecture. For all $r\ge2$, if $6\mid K_r$, then

$$K_r\equiv 6+6^2(r-2)\pmod{6^2\cdot 8}.$$

Equivalently, if $K_r>2$, then $K_r$ is a multiple of $6$ and

$$\frac{K_r-6}{6^2}\equiv r-2\pmod{8}.$$

The statement is proposed as a strengthening of a previously established congruence for all primary pseudoperfect numbers divisible by $6$, including any with more than eight prime factors. No resolution is supplied in the source.