Reciprocal polynomials with all zeros on the unit circle

Let f(x)=adxd+ad−1xd−1+⋅⋅⋅+a0∈ℝ[x] be a reciprocal polynomial of degree d. We prove that if the coefficient vector (ad,ad−1,…,a0) or (ad−1,ad−2,…,a1) is close enough, in the l1-distance, to the constant vector (b,b,…,b)∈ℝd+1 or ℝd−1, then all of its zeros have moduli 1.