Primitive non-deficient number component conjecture

Let $N$ be a primitive non-deficient number with

$$N=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k},$$

where $p_1<p_2<\cdots<p_k$ are distinct primes. Set

$$R=R(N)=p_1p_2\cdots p_k.$$

Primitive non-deficient number component conjecture. There is an $i$, with $1\leq i\leq k$, such that

$$p_i^{a_i+1}<kR.$$

The conjecture asks for a bound analogous to the paper's result for odd perfect numbers. The source presents it as plausible, but no resolution is given here; the paper proves the stronger inequality in some cases.