  An on-going challenge within scalable optical quantum information processing
is to increase the collection efficiency $\varepsilon$ and the photon
indistinguishability $\eta$ of the single-photon source towards unity. Within
quantum dot-based sources, the prospect of increasing the product $\varepsilon
\eta$ arbitrarily close to unity was recently questioned. In this work, we
discuss the influence of the trade-off between efficiency and
indistinguishability in the presence of phonon-induced decoherence, and we show
that the photonic "hourglass" design allows for improving $\varepsilon \eta$
beyond the predicted maximum for the standard micropillar design subject to
this trade-off. This circumvention of the trade-off is possible thanks to
control of the spontaneous emission into background radiation modes, and our
work highlights the importance of engineering of the background emission in
future pursuits of near-unity performance of quantum dot single-photon sources.
