Each $3$-polytope with minimum degree $5$ has a $7$-cycle with maximum degree at most $15$

Let $\varphi_P(C_7)$ ($\varphi_T(C_7)$) be the minimum integer $k$ with the property that each $3$-polytope (respectively, each plane triangulation) with minimum degree $5$ has a $7$-cycle with all vertices of degree at most $k$. In 1999, Jendrol', Madaras, Soták, and Tuza proved that $15\le\varphi_T(C_7)\le17$. It is also known due to Madaras, Škrekovski, and Voss (2007) that $\varphi_P(C_7)\le359$.
We prove that $\varphi_P(C_7)=\varphi_T(C_7)=15$, which answers a question of Jendrol' et al. (1999).