Assuming large cardinal axioms, which real numbers are in $L[\mathrm{cf}]$, where $\mathrm{cf}$ is the cofinality function on ordinals?

$L[\mathrm{cf}]$ is the minimal inner model that 'knows' the cofinality of every ordinal in V, or more precisely, $∀α\,\{(β,\mathrm{cf}(β)):β<α\}∈L[\mathrm{cf}]$. Cofinality in $L[\mathrm{cf}]$ need not equal cofinality in $V$.

I have a solution to a simpler problem:<br/>
<i>Theorem:</i> A real number is in $L$[Card] iff it is in the minimal inner model with a proper class of measurable cardinals (assuming the sharp for this model exists).  Here, Card is the cardinality function.<br/>
<i>Corollary</i> (under the same assumption): The theory of $(L[\mathrm{Card}],∈,\mathrm{Card})$ is $Δ^1_3$.<br/>
<i>Proof sketch of the theorem:</i>  Starting with the minimal inner model $M$ with a proper class of measurable cardinals, iterate the first measurable cardinal until it becomes $ω_1^V$, the second until it becomes $ω_2^V$, and so on for every regular successor cardinal in $V$.  For the converse, every $ω_{α+ω}$ can be shown to be measurable in HOD<sup><i>L</i>[Card]</sup> with $\{ω_{α+1}, ω_{α+2}, ω_{α+3}, ...\}$ Prikry generic over HOD<sup><i>L</i>[Card]</sup>.

For the cofinality problem, my conjecture is to use a model $M$ (closely related to $L[\mathrm{cf}]$) such that:<br/>
nonmeasurable regular cardinals in $M$ have $V$-cofinality $ω$,<br/>
measurable cardinals of order $α$ in $M$ have $V$-cofinality $ω_{α+1}$ (or $ω_α$ if $α$ is weakly inaccessible (in $V$) plus a finite ordinal),<br/>
and $M$ is obtained by iterating away the least mouse with a measure concentrating on cardinals with $o(κ)=κ$, and iterating the measures until the above correspondences hold.  However, I do not know whether this works, or for that matter, whether the theory of $L[\mathrm{cf}]$ is generically absolute.

An extension of the problem is to set $\mathrm{cf}(κ)=κ+α$ iff $κ$ is weakly $α$-Mahlo, which is well-defined for $α≤κ^+$.  A $κ^+$-Mahlo $κ$ is called greatly Mahlo.  For comparison, an ordinal has cofinality $ω_{α+2}$ iff ordinals of cofinality $ω_{α+1}$ are stationary below it.  (Also, if singular $κ$ pose a problem, I would be happy to see a solution for this extension with $\mathrm{cf}(κ)$ restricted to regular $κ$.) It appears plausible that the construction using orders of measurability can be extended to this problem, with weakly greatly Mahlo cardinals corresponding to $M$-cardinals that are strong up to a measurable.  If that is the case, then under large cardinal axioms, all reals in $L[\mathrm{cf}]$ are still $Δ^1_3$.