Hi Pierre, "fully believable Beethoven" won't happen. However, "able to fool the general public or reporters Beethoven" has already been done. I did my PhD in Algorithmic Composition. Even David Cope who has made a career out of promoting these sorts of ideas (see Experiments in Musical Intelligence) has not addressed what is called the Theory Problem, viz:
Composition is more than an application of music theory, as a creative process it breaks rules and makes rules at will to suit its needs. Computers cannot do that yet.
Music theory is created by analysing a bunch of compositions and finding common rules, ignoring the anomolies, thus it is a reductive process. Composition is an expansive process, and it a computer applies music theory rules to create music, then the result is average, because of the averaging of the rule set. The way to overcome this is to use controlled randomness, a la Xenakis or Koenig, to create compelling and sophisticated compositions.
The first book on algorithmic composition (Experimental Music by Hiller and Isaacson) made this observation in 1959. Modern attempts at using artificial intelligence have so far not advanced the art beyond controlled randomness. Some of the best examples of "style replication" come from using Markov chains for conditional generation, the way this works is to take a composition by, say, Beethoven and use random jumps into it such that (this is a second-order Markov chan example), I already have the notes "C" and "F", then I jump randomly into the Beethoven piece, and then step through it until I find a "C" followed by an "F", and I take the next note as the note I'm looking for in my piece. This will create a fairly convincing replication of Beethoven's style.
Google has been playing with its AI engine in this regard, but so far the results that I've heard are no better than Markov chains, although the way of getting to that result is fundamentally different.
I hope this helps!