Is the following statement consistent:

>``**There is no non-trivial c.c.c forcing notion adding a minimal generic real**''?

The question is related to Prikry's question:  Is it consistent that any non-trivial c.c.c forcing notion adds a Cohen real or a Random real?

See also [https://mathoverflow.net/questions/149316/a-special-c-c-c-forcing-notion-and-adding-minimal-generic-reals](https://mathoverflow.net/questions/149316/a-special-c-c-c-forcing-notion-and-adding-minimal-generic-reals) where it is shown that the answer to the above question is no, if we assume $CH$ or $MA+\neg CH$ (this follows from the results of Judah-Shelah in _[Forcing Minimal Degree of Constructibility](https://doi.org/10.2307/2275046)_ ([JSTOR](https://www.jstor.org/stable/2275046), [free author version](https://people.math.wisc.edu/~miller/old/m873-09/bib/judah-minimal.pdf))) Also it is clear that, a model for Prikry's question is also a model for the above mentioned question.

**Remark.** By a theorem of Shelah, a modified version of Prikry’s conjecture holds in the
context of Souslin c.c.c forcing: i.e. every Souslin c.c.c forcing either adds a
Cohen real or is a Maharam algebra. Indeed, Shelah showed that any Souslin
c.c.c forcing which is not $ω^ω$-bounding adds a Cohen real. See

S. Shelah, How special are Cohen and random forcing. Israel Journal
of Math. 88 (1-3), pp. 159-174, (1994), doi:[10.1007/BF02937509](https://doi.org/10.1007/BF02937509), arXiv:[math/9303208](https://arxiv.org/abs/math/9303208), [author pdf](https://shelah.logic.at/files/95834/480.pdf)