Bernoulli-denominator conjecture for Toda-prime sets

For an even integer $2m$, let $P(2m)$ be the set of primes $p$ such that $p-1\mid 2m$, so that

$$D_{2m}=\prod_{p\in P(2m)}p,$$

where $D_{2m}$ is the denominator of the Bernoulli number $B_{2m}$. Let $F(d)=\min\{2m>0:D_{2m}=d\}$. Also let $T(n)$ be the set of Toda primes of $n$, and write $t(n)=|T(n)|$. Bernoulli-denominator conjecture. Let $d$ be a Bernoulli denominator.

(i) If $F(d)=4a$ for some integer $a$, then $T(am)=T(a)$ whenever $D_{4am}=D_{4a}$.

(ii) If $D_{4am}\neq D_{4a}=d$, then $t(am)\geq t(a)$.

The conjecture proposes that equality or growth of Bernoulli denominators controls the Toda-prime sets and their cardinalities. It extends several case-by-case propositions proved or stated earlier in the paper; its general validity is left open.