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The ground-state hyperfine splitting in muonic hydrogen provides direct sensitivity to the proton Zemach radius, reflecting the combined effects of its spatial charge and magnetic moment distributions. Despite its fundamental importance, the hyperfine spin-flip transition has not yet been observed, owing to its extremely small excitation probability and the weak de-excitation energy underlying the measurement, making its experimental investigation particularly challenging. The FAMU experiment aims to achieve this goal by inducing the spin-flip transition with a pulsed mid-infrared laser in a cryogenic muonic-hydrogen target doped with a small fraction of oxygen. The hyperfine structure, if excited by the laser beam, should produce a measurable variation of the characteristic X-ray emission spectrum of muonic oxygen, which thus may provide a distinct and measurable signal of the transition. The laser system combines a 1064 nm Nd:YAG laser with a tunable Cr:forsterite laser centered at 1262 nm through difference-frequency generation in a BaGa₄Se₇ nonlinear crystal. An active control system ensures a frequency stability of 15 MHz (2.3 pm) over a 24-hour period. The resulting radiation at 6780 nm features a linewidth of about 100 MHz (15 pm) and a maximum pulse energy of more than 5 mJ. Six measurement campaigns have been conducted at the ISIS Muon Source at the Rutherford Appleton Laboratory (UK). The laser system operated continuously for more than 10 consecutive days in each campaign, showing its reliability and capability for precision hyperfine spectroscopy. These results represent a crucial step toward the first direct observation of the muonic hydrogen hyperfine transition.