Non-invertibility of RSA for two users
What you're asking for has been called "correlated product" security for a trapdoor function.  See this paper by Rosen and Segev, which lays out the definition.  Your question fixes $k$ from their Definition 3.2 to $2$;  as you (and their paper) note, we know that the RSA trapdoor function with exponent $e\leq k$ is not correlated-product secure.

I don't know how to give a reduction to plain RSA to prove what you want, and I'd guess that it's very hard.  The best I can think of is to try to prove this under the so-called Phi-hiding assumption (see Cachin, Micali, Stadler, Eurocrypt'99, or reference below).  This may follow by combining Theorem 3.3 of 1 (which says that a lossy trapdoor function is correlated-product secure) with Section 5 of this this paper by Kiltz, O'Neill, and Smith that shows the RSA function is lossy under that assumption.  I haven't checked that the parameters works out, though.