The shifted-prime largest-prime-factor extremes conjecture

Let $a$ be the fixed shift appearing in the statement, let $P^{+}(m)$ denote the largest prime factor of $m$, and let $\boldsymbol{\vartheta}$ and $\boldsymbol{\Theta}$ be real numbers. The shifted-prime largest-prime-factor extremes conjecture. For any $0<\boldsymbol{\vartheta},\boldsymbol{\Theta}<1$, there are infinitely many primes $p$ with

$$P^{+}(p+a)<p^{\boldsymbol{\vartheta}}$$

and infinitely many primes $p$ with

$$P^{+}(p+a)>p^{\boldsymbol{\Theta}}.$$

This is presented as a special version of the preceding distribution conjecture. The source does not state that either assertion has been proved in the full range $0<\boldsymbol{\vartheta},\boldsymbol{\Theta}<1$.