The two-term supercongruence conjecture for the original sporadic sequences

Let $A(n)$ be one of the 15 original sporadic sequences. For a prime $p\geq 5$ and integers $m,r\geq 1$, write

$$A(mp^r) \equiv A(mp^{r-1}) \pmod{p^{\lambda r}}.$$

Here $\lambda=3$ except for the sequences $\bf{B}$, $\bf{C}$, $\bf{E}$, $\bf{F}$ and $s_{18}$, for which $\lambda=2$.

Two-term supercongruence conjecture. For all primes $p\geq 5$ and all integers $m,r\geq 1$,

$$A(mp^r) \equiv A(mp^{r-1}) \pmod{p^{\lambda r}}.$$

The conjecture concerns the arithmetic properties of the 15 sporadic sequences, which arise from three-term recurrences and have modular parametrizations, binomial-sum representations and geometric origins. The paper states that this is the original conjecture and proves the last remaining case, so the claim is solved.