Consider the normal modal logic system $\mathbf{TAR1}$ given by $\mathbf{T}$ plus the following axiom:

$$\mathrm{AR1}: \lozenge \square p \rightarrow (\square p \lor \square (p \rightarrow \square p))$$

Analysis using normal forms shows that it is included in $\mathbf{S4.4}$. However, algebraically I cannot either prove or disprove that it is strictly weaker. I was wondering is someone has a proof or an argument based on Kripke semantics.

Additional info, in case it helps:

$\mathbf{S4.4}$ is usually defined as $\mathbf{S4}$ plus axiom $\mathrm{R1}$: $$\mbox{R1}: \lozenge \square p \rightarrow (p \rightarrow \square p)$$ I find that $\mathbf{S4.4}$ is also $\mathbf{T}$ plus the following axiom: $$\mathrm{4.4}: \lozenge \square p \rightarrow \square (p \rightarrow \square p)$$

I also find that $\mathbf{TAR1}$ is equivalent to $\mathbf{TR1}$, which is $\mathbf{T}$ plus $\mathrm{R1}$. But the question now becomes whether $\mathbf{TR1}$ is equivalent to $\mathbf{S4.4}$.

Finally, I am trying to prove $\mathrm{4.4}$ in $\mathbf{TAR1}$, but it does not seem to be possible using only normal forms of degree $\le 3$. So, unless my calculations are wrong, either the proof involves modalities of degree 4, or $\mathbf{TAR1}$ is a distinct system. If distinct, then it would not include $\mathbf{S4}$.

Philosophically, this particular system may not be particularly relevant; but an answer would help me the algebraic properties of other similar systems that I'm analyzing.