Criterion for the algebraic independence of values of hypergeometric $E$-functions (even case)

For the hypergeometric function
\begin{gather*} \varphi_{\overline\lambda}(z)=\sum_{n=0}^\infty\frac 1{(\lambda_1+1)_n\dotsb(\lambda_t+1)_n}\Bigl(\frac zt\Bigr)^{tn}, \qquad \overline\lambda=(\lambda_1,\dots,\lambda_t), \\ \lambda_j\in\mathbb Q\setminus\{-1,-2,\dots\}, \qquad j=1,\dots,t, \end{gather*}
satisfying a linear differential equation of order $t$, for the case of an event prime to 3, a criterion is obtained for the algebraic independence over $\mathbb Q$ of the numbers $\varphi_{\overline\lambda}^{(k)}(\alpha)$, $k=0,1,\dots,t-1$, where $\alpha\in\mathbb A\setminus\{0\}$. The case of odd $t$ was fully investigated in the author's previous papers.