If we start with $V\models\lozenge$, it is not hard to force the failure of diamond. You can blow up the continuum, or destroy all the Suslin trees. You can blow up the continuum of $\aleph_1$, and then collapse $\aleph_1$ to be countable.
There are many ways of doing that, but all of them (that I could think of, with the help of a few people over the day) include one of the two:
- Blowing up the continuum,
- Collapsing cardinals.
Is it consistent that $V\models\lozenge$, and $r$ is a $V$-generic real such that $V[r]\models\lnot\lozenge+\sf CH$ and no cardinals were collapsed between $V$ and $V[r]$?
If the answer is positive, can we strengthen the preservation of $\sf CH$ by requiring also that the continuum function remains the same (so no blowing up power sets of larger cardinals somehow)?