Notations: $R$- Noetherian graded ring and $I,J$ homogeneous ideals in $R$

**Definition:**
The projective dimension of $R/I$, denoted $pd(R/I)$, is the length of a minimal free graded resolution of $R/I$:
$0 \rightarrow \bigoplus_{j} R(-j)^ {\beta_{p,j}(R/I)}\rightarrow \bigoplus_{j} R(-j)^ {\beta_{p-1,j}(R/I)} \rightarrow \cdots \rightarrow \bigoplus_{j} R(-j)^ {\beta_{0,j}(R/I)} \rightarrow
R/I \rightarrow 0$.

where $p \leq n$ and $R(-j)$ indicates the ring $R$ with the shifted graduation such that, for all $a \in \mathbb{Z}$, $R(-j)_{a}=R_{a-j}$ and $\beta_{i,j}(R/I)$ is the number of copies of $R(-j)$ appearing in the $i$-th module of the resolution, and is called graded Betti number degree $(i,j)$. The $i$-th Betti number is $\beta_{i}(R/I)=\sum \limits_{j \in \mathbb{Z}}\beta_{i,j}(R/I)$. The Castelnuovo-Mumford regularity (or simply regularity) of $R/I$ is

$\operatorname{reg}(R/I)=\max\{j-i\mid\beta_{i,j}(R/I)\neq 0\}$

Let $I$ be quadratic square free monomial ideal in $k[x_1,\ldots,x_n]$ and $J$ be quadratic square free monomial ideal in $k[y_1,\ldots,y_m]$ with $I\ncong J$. Suppose $\beta_{i,2i}(I) \neq 0$ and $\beta_{j,2j}(J) \neq 0$ for some $i,j$. Let $IJ$ be square free monomial in $k[x_1,\ldots,x_n,y_1,\ldots,y_m]$.

Can we say that $\beta_{(i+j+2),2(i+j+2)}(IJ)\neq 0$ ?