  The main goal of the paper is to provide a quantitative lower bound greater
than $1$ for the relative projection constant $\lambda(Y, X)$, where $X$ is a
subspace of $\ell_{2p}^m$ space and $Y \subset X$ is an arbitrary hyperplane.
As a consequence, we establish that for every integer $n \geq 4$ there exists
an $n$-dimensional normed space $X$ such that for an every hyperplane $Y$ and
every projection $P:X \to Y$ the inequality $||P|| > 1 + \left (8 \left ( n + 3
\right )^{5} \right )^{-30(n+3)^2}$ holds. This gives a non-trivial lower bound
in a variation of problem proposed by Bosznay and Garay in $1986$.
