Sibling-pair and sibling-tuple conjecture for groups of orders $p^4$ and $p^5$

Let $p>3$ be prime. A sibling-$l$-tuple is a collection of $l$ groups identified as siblings under the group identification framework of the paper.

Sibling-pair and sibling-tuple conjecture. Among the $15$ groups of order $p^4$, there is exactly one sibling-pair. Among the $2p+2\gcd(p-1,3)+\gcd(p-1,4)+61$ groups of order $p^5$, there are two sibling-tuples of length $l=(p+1)/2$, two sibling-tuples of length $l=(p-1)/2$, one sibling-tuple of length $l=\gcd(p-1,4)$, one sibling-tuple of length $l=2$, and, if $\gcd(p-1,3)=3$, two sibling-tuples of length $l=3$.

The conjecture is motivated by computational evidence for groups of small prime-power order, including the observed sibling configurations for groups of orders $3^5$ and $3^6$. Its status is not established by the supplied source context.