39 consecutive same bases is sure to be polymorphic if derived from animal or plant tissue. Apart from this CE is usually carried out at high temperature and in the presence of deionised formamide both of which will minimise secondary structure issues so I would expect this to run normally. If you were sequencing this template then you will expect the sequencing peak heights to fall markedly across the 39bases due ,in part, to reagent depletion

Long homopolymer regions of A or T can cause problems during capillary electrophoresis due to "polymerase slippage" in the repeating region. I would use your standard primers first and then try an anchored primer if you see significant noise after the repeating T region. An anchored primer has a run of T's and an "anchoring" base of G, A, or C at the 3' end. The 3´ base anchors the primer into place at the end of the homopolymer region.