Block branching Miller forcing and covering numbers for prediction

We call a function from ω<ω to ω a predictor. A predictor π predicts f∈ωω constantly if there is n<ω such that for all i<ω there is j∈[i,i+n) with f(j)=π(f↾j). θω is the smallest size of a set P of predictors such that every f∈ωω is constantly predicted by some predictor in P. θubd is the smallest cardinal κ satisfying the following: For every b∈ωω there is a set P of predictors of size κ such that every f∈∏n<ωb(n) is constantly predicted by some predictor in P. We prove that θubd is consistently smaller than θω.