Csdrhrt

Suppose $A$ is an algebra over $mathbb{C}$. Let $M$ and $N$ be $A$-modules with $Ext^{1}(M,N) neq 0 neq Ext^{1}(N,M)$.
By $Ein Ext^1(N,M)$, I mean we have a short exact sequence of the form
$$ 0 rightarrow M rightarrow E rightarrow N rightarrow 0.$$
I am wondering how the following two sets are related:
$$S_1 := { d : M oplus N textrm{ has a submodule of dimension } d } $$
$$S_2 := { d : exists textrm { nonsplit } Ein Ext^1(M,N) textrm { or } Ein Ext^1(N,M) textrm { s.t. } E textrm { has a submodule of dimension } d }.$$
I think $S_2 subseteq S_1$ since if $Ssubseteq E$ is a submodule with $0 rightarrow M rightarrow E rightarrow N rightarrow 0$, then viewing $M$ as a submodule of $E$, consider the exact sequence
$$0 rightarrow Scap M xrightarrow {i} S rightarrow textrm{coker }i rightarrow 0.$$
Then if we define a map $varphi :$ coker $i cong S/(Scap M)rightarrow N cong E/M$ by $[s] mapsto [s]$, this will be injective so the module $(S cap M)oplus textrm{im } varphi$ will be a submodule of $Moplus N$ with the same dimension as $S$.
I also think $S_2 subseteq S_1$ but I haven't been able to think of any way to see this.

Would it make a difference if I also assumed that

1) $M$ and $N$ are indecomposable,

2) $dim Ext^1(M,N) = dim Ext^1 (N,M) = 1$,

3) The nonsplit extensions $0 rightarrow M xrightarrow{f_1} E_1 rightarrow N rightarrow 0$ and $0rightarrow N xrightarrow {f_2} E_2 rightarrow M rightarrow 0$ have the property that $f_1$ and $f_2$ are minimal left approximations?