It is known that given $M\subseteq N$ structure and a substructure such that $M\equiv N$, we do not necessarily have $M\prec N$.

Similarly we can have $M\_0\subseteq M\_1\subseteq M\_2$ such that $M\_0\equiv M\_1\equiv M\_2$ that are not elementary substructure to one another.

Is there a chain where this stops to be true? For example, if I have $\kappa$ that is measurable such that $(M\_\mu\mid\mu\in(\kappa\cap Card))$ chain of substructures, all with the same theory $T$ (for convenient sake, let's fix some countable language), and (to have somewhat tame behavior) all are bounded subsets of $\kappa$, does there exists cardinals $\alpha<\beta<\kappa$ such that $M\_\alpha\prec M\_\beta$ (I would imagine that if 1 such pair exists, there will reflect downwards and we will have at least stationary chain of elementary substructures)?

I played a bit with the problem and I think that the answer is yes. Let $\kappa$ be measurable, and $\mathcal U$ be $\kappa$-complete normal non-principal ultrafilter on $\kappa$, and $j$ be the induced elementary embedding on it's ultrapower.

First let's assume that $|M\_\lambda|\ge\lambda$ for $\cal U$-large set $X$, as if $|M\_\lambda|<\lambda$ for $\cal U$-large set then it is trivial by normality and the fact that $\kappa$ is a strong limit.

Let $Y=\{\mu\mid \text{there exists $\alpha\in X$ such that }\mu=|M\_\mu|\}$ and for $i\in Y$ let $Y\_i$ be first model of size $i$, $Y$ satisfy "for all $\mu$ we have $\mu\in Y\iff (\mu\in X\land \mu\text{ is larger than every $X\_i$ for $i<\mu$})$". Because $\kappa\in j(X)$, the ultrapower computers the $X\_i$ for $i<\kappa$ correctly and the ultrapower thinks $X\_{\kappa}\ge\kappa$, then $\kappa\in j(Y)$.

In summery we have $Y\subseteq \kappa$ a $\cal U$-large set such that $(Y\_i\mid i\in Y)$ is a chain of elementary equivalent substructures such that $|Y\_i|=i$.

My next step was to try to reflect downwards some properties on $Y$ to somehow get a fixed point on "approximate"-Skolem hull operation. Basically fix some Mahlo, or limit of Mahlos, $\nu\in Y$ and look at $(Hull\_{Y\_{\nu}}(Y\_\mu)\mid \mu\in Y\cap\nu)$ and hope it will somehow lend on some $Y\_\mu\subseteq Y\_\nu$, but I couldn't really find a way to do this.

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Does what I wrote can get me somewhere? If it is indeed true, is there some tighter conditions (as my idea was basically to reflect down to $Y$ all of the LCA I want, so it is obviously an overkill if this method indeed works)?