The following question was asked years ago on MSE, but let me recap it:

Question: Is there anything currently known about the exact consistency strength of "$\mathsf{ZF}$ + both $\omega_1$ and $\omega_2$ are singular?" Or could we find a better bound?

It is well-known that if both $\omega_1$ and $\omega_2$ are singular, (of course, without choice) then $0^\sharp$ exists. It is quite famous that we can make all uncountable cardinals singular from a proper class of strongly compact cardinals. However, if we only consider $\omega_1$ and $\omega_2$ then Gitik's result is overkill: Apter proved in

Arthur W. Apter.

$\mathsf{AD}$ and patterns of singular cardinals below $\Theta$.J. Symbolic Logic61 (1996), no. 1, 225–235.

that if we start from $L(\mathbb{R})\models\mathsf{ZF+AD+DC}$ then we can find an extension $N$ of $V$ whose cardinals are the same with that of $V$, and both of $\omega_1$ and $\omega_2$ have cofinality $\omega$. Thus we have "$\mathsf{ZFC}$ + there are $\omega$ many Woodin cardinals" as an upper bound.

(Apter's result for $\omega_1$ and $\omega_2$ is further generalized in a paper by Apter, Jackson, and Löwe. Also, note that what Apter proved is stronger than I stated: he actually proved that if we work over $V=L(\mathbb{R})\models\mathsf{ZF+AD+DC}$ and if $A$, $B$ are sets partitioning the set of regular cardinals below $\Theta^{L(\mathbb{R})}$ then there is an extension $N$ of $V$ such that cardinals in $V$ and those in $N$ are the same and cardinals in $A$ has cofinality $\omega$, but cardinals in $B$ are still regular in $N$.)

On the other hand, Schindler proved in

Ralf-Dieter Schindler,

Successive weakly compact or singular cardinals.J. Symbolic Logic64 (1999), no. 1, 139–146.

that if $\kappa$ is measurable in $\mathsf{HOD}$ and inaccessible in $V\models \mathsf{ZF}$ and there is $\delta<\kappa$ such that $\delta^+<\kappa$ and both of $\delta$ and $\delta^+$ are singular, then there is an inner model of a Woodin cardinal.

It brings possibly related questions that are also curious for me: First, it is easy to see that if we start from $\mathsf{ZF}$ with both $\omega_2$ and $\omega_3$ are singular, then there is a generic extension of $V$ that thinks $\omega_1$ and $\omega_2$ are singular.

Question.How about the opposite direction? So, can we derive the consistency of "$\mathsf{ZF}$ + both $\omega_2$ and $\omega_3$ are singular" from that of "$\mathsf{ZF}$ + both $\omega_1$ and $\omega_2$ are singular"?

Regarding the upper bound, the role of $\mathsf{AD+DC}$ in Apter's proof is a normal measure over $\omega_1$ and $\omega_2$. He then uses a product of Prikry forcings and constructs an inner model of a generic extension. It seems like providing normal measures is the only role of $\mathsf{AD}$.

Question.Can we find a model of $\mathsf{ZF}$ + "both of $\omega_1$ and $\omega_2$ have a normal measure" from large cardinal hypotheses weaker than $\omega$ many Woodin cardinals?

(The paper by Apter, Jackson, and Löwe pointed out in Theorem 32 that if we start from $\mathsf{ZFC}$ two measurable cardinals then we have an extension $N\models\mathsf{ZF}$ of $V$ that thinks $\omega_1$ and $\omega_3$ are measurable and $\omega_2$ is regular. I have no idea whether it is related to an answer to the above question, and probably not.)