A damped oscillator heat bath model is a modification of the standard heat bath model, wherein each bath oscillator itself has a Markovian coupling to its own heat bath [A. V. Plyukhin, Phys. Rev. E 99, 052125 (2019)]. We modify such a model to one where the resulting damping of the oscillators is linear in their frequency rather than being a constant. We find that this generates a memory kernel which behaves like 𝑘⁡(𝑡)∼1/𝑡 as 𝑡→∞, which is a boundary case not considered in previous works. As the memory kernel does not have a finite integral, the reduced system is subdiffusive, and we numerically show that diffusion goes as ⟨Δ⁢𝑄2⁡(𝑡)⟩∼𝑡/ln⁡(𝑡) as 𝑡→∞. We also numerically calculate the velocity correlation function in the asymptotic regime and use it to confirm the aforementioned subdiffusion.