The non-occurrence of $\zeta(2)$ as a $\phi^4$ period

Let $\mathcal{P}_{\phi^4}$ denote the space of $\phi^4$ periods and let $\zeta(2)$ be the Riemann zeta value at $2$. Non-occurrence conjecture.

$$\zeta(2)\notin\mathcal{P}_{\phi^4}.$$

No $\phi^4$ graph is known to evaluate to $\zeta(2)$; the stronger expectation involving logarithmic and generalized Feynman periods is stated separately in the source.