Mais is fine here but you can drop it too. You might want to start a new sentence in that case or replace the comma by a semicolon though.
Volkswagen peut tenir toutes les conférences de presse qu’elle veut (or qu’il veut); ses excuses à la noix de dernière minute n’ont guère de valeur aux yeux des consommateurs.
“Mais” is like “but”: it is a mark of opposition between the implications of the two parts of the sentence:
Je n’aime pas les légumes, mais comme c’est toi qui les a préparés je vais y goûter. / I don’t like vegetables, but as you cooked them yourself I will give them a chance.
The first part explain that in a normal case the speaker would not eat these vegetables, the second one explains that the first part is not precisely exact, in particular for the present situation.
In your example there is no such contradiction, thus the use of “mais” is not correct. The sentence must be written without it. A correct construction with “mais” could be:
Volkswagen peut tenir toutes les conférences de presse qu’elle veut, mais ne peut pas redonner confiance aux consommateurs.
Here there is an opposition between the first part (Wolkswagen can do things as they please) and the last part (Wolkswagen can’t do the important things we are actually talking about). Note that it has a different implicit meaning than what you wanted to say:
In your example it is implied that usually press conferences are a way to win the trust of consumers (but in this case it won’t be enough).
In my last example there is no such implication. The two things are considered different ones without any link.