  We study bounds on ratios of fluctuations in steady-state time-reversal heat
engines controlled by multi affinities. In the linear response regime, we prove
that the relative fluctuations (precision) of the output current (power) is
always lower-bounded by the relative fluctuations of the input current (heat
current absorbed from the hot bath). As a consequence, the ratio between the
fluctuations of the output and input currents are bounded both from above and
below, where the lower (upper) bound is determined by the square of the
averaged efficiency (square of the Carnot efficiency) of the engine. The
saturation of the lower bound is achieved in the tight-coupling limit when the
determinant of the Onsager response matrix vanishes. Our analysis can be
applied to different operational regimes, including engines, refrigerators, and
heat pumps. We illustrate our findings in two types of continuous engines:
two-terminal coherent thermoelectric junctions and three-terminal quantum
absorption refrigerators. Numerical simulations in the far-from-equilibrium
regime suggest that these bounds apply more broadly, beyond linear response.
