Variant fails with private default constructor

[dmenendez-gruposantander]

Boost.Variant fails if its types have a private default constructor, using gcc 12.2
Minimal example: https://godbolt.org/z/cG4PnW8ee

#include <boost/variant.hpp>
#include <boost/variant2.hpp>
class Foo
{
    public:
        Foo(int a_) : a(a_) {}
        int a;
    private:
        Foo();
};
struct Voo
{
    Foo f;
    double d;
};
struct Doo
{
    Foo f;
    int s;
};
#ifndef USE_VARIANT2
    using VooDoo = boost::variant<Voo, Doo>;
    void testVooDoo(const VooDoo& voodoo)
    {
        VooDoo v{Voo{Foo(3)}};
    } 
#else
    using VooDoo = boost::variant2::variant<Voo, Doo>;
    void testVooDoo(const VooDoo& voodoo)
    {
        VooDoo v{Voo{Foo(3)}};
    }
#endif

If I change to use Boost.Variant2, it works.
If I go back to gcc 7.5, it works (fails after that version).

The error shows:

/opt/compiler-explorer/libs/boost_1_80_0/boost/variant/variant.hpp:194:17:   required from 'struct boost::detail::variant::find_fallback_type<boost::mpl::l_item<mpl_::long_<2>, Voo, boost::mpl::l_item<mpl_::long_<1>, Doo, boost::mpl::l_end> > >'
/opt/compiler-explorer/libs/boost_1_80_0/boost/variant/variant.hpp:1252:17:   required from 'class boost::variant<Voo, Doo>'
<source>:26:16:   required from here
/opt/compiler-explorer/libs/boost_1_80_0/boost/type_traits/has_nothrow_constructor.hpp:27:84: error: 'Foo::Foo()' is private within this context
   27 | template <class T> struct has_nothrow_constructor : public integral_constant<bool, BOOST_HAS_NOTHROW_CONSTRUCTOR(T)>{};
      |                                                                                    ^~~~~~~~~~~~~~~~~~~~~~~~~~~~~
<source>:9:9: note: declared private here
    9 |         Foo();
      |         ^~~

It mentions find_fallback_type, so I guess that at some point it tries to determine the "fallback type" to default construct the VooDoo.
However, nowhere in my program I need to default construct a VooDoo, so I'm not sure why that type needs to be "computed".

Also, somehow Boost.Variant2 doesn't fail, so there must be a way to make this work.