there is no clear evidence of that - and in case - Bernadotte was not able to give it.
due to two facts

1. He did not know the overall situation - so it could be that Davout engaged a Prussian minor force - one has to be always bear in mind, that neither Bernadotte, nor Davout, nor Napoleon did know that Davout would fight by accident the Prussian main force.

2. Bernadotte was at that time well on the road to Apolda and some of his units already where passing the Saale at Dornburg and where stuck there due to the narrow roads and the steep ascent.

In case Bernadotte would give help, he would risk (not knowing where the Prussian main force was) to jeopardize Napoleons planning and he would be late to support Davout anyway.

It could well have happened - in case he chose to do so, that he would be like Drouet in 1815 (and strangly nobody is accusing this general of being a traitor or not supporting).

It is quite important to take the position of Bernadotte's units into account.

In case you have abit time - read Sir Plunket Barton, who sums it up nicely.