This is a generalization of this question.

Let $P_1, \ldots, P_m$, $Q_1, \ldots, Q_k \in \mathbb{C}[x_0,\ldots,x_n]$ be linear homogenous polynomials. Let $f_1, \ldots, f_s$ be a homogenous quadratic irreducible polynomials of degree $2$.

Assume that for every $i$ and for every $j$ the ideal $\langle P_i, Q_j \rangle$ contains some $f_l$.

Assume also that the dimenstion of the span of $\{f_1, \ldots, f_s \}$ (in the vector space of all quadratic homogenous polynomials in $\mathbb{C}[x_0,\ldots,x_n]$) is equal to some constant $c$ .

**Question:** Is it true that the dimension of the span of $\{P_1, \ldots, P_m \}$ or the dimension of the span of $\{Q_1, \ldots Q_k \}$ is less than some constant (i.e. some function from $c$)?

I can affirmatively answer this question if $s$ (the number of quadratic polynomials) is bounded by a constant:

Consider those polynomials in $\{f_1, \ldots, f_s\}$ that belongs to $\langle P_1, Q_j \rangle$ for some $j$. W.l.o.g. we can assume that this set is $\{f_1, f_2,, \ldots, f_{s'} \}$ for some $s' \le s$.

Consider $M_i:= f_i \cap P_1$ (I mean the intersection of the zeros $f_i$ and $P_1$) for some $i \le s'$.

This set is the zeros of a quadratic form in plane $P_1$ with codimension $1$ (it can not be $P_1$ since $f$ is irreducible). For some $j$ the intersection $Q_j \cap f$ must contain a subspace of codimension $2$. Hence $M_i$ is the union of one or two subspace of codimension $2$. So, there exists at most $2s'$ subspaces of codimension $2$ such that every $Q_j$ must contain at least one of them. Now, it is not difficult to see that the dimension of the span of $\{Q_1, \ldots, Q_k\}$ is bounded by $4s' \le 4 s$. The similar argument works for the dimension of the span of $\{P_1, \ldots, P_m\}$.