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We report the discovery and characterisation of a pair of sub-Neptunestransiting the bright K-dwarf TOI-1064 (TIC 79748331), initially detected inTESS photometry. To characterise the system, we performed and retrieved CHEOPS,TESS, and ground-based photometry, HARPS high-resolution spectroscopy, andGemini speckle imaging. We characterise the host star and determine $T_{\rmeff, \star}=4734\pm67$ K, $R_{\star}=0.726\pm0.007$ $R_{\odot}$, and$M_{\star}=0.748\pm0.032$ $M_{\odot}$. We present a novel detrending methodbased on PSF shape-change modelling and demonstrate its suitability to correctflux variations in CHEOPS data. We confirm the planetary nature of both bodiesand find that TOI-1064 b has an orbital period of $P_{\rm b}=6.44387\pm0.00003$d, a radius of $R_{\rm b}=2.59\pm0.04$ $R_{\oplus}$, and a mass of $M_{\rmb}=13.5_{-1.8}^{+1.7}$ $M_{\oplus}$, whilst TOI-1064 c has an orbital period of$P_{\rm c}=12.22657^{+0.00005}_{-0.00004}$ d, a radius of $R_{\rmc}=2.65\pm0.04$ $R_{\oplus}$, and a 3$\sigma$ upper mass limit of 8.5 ${\rmM_{\oplus}}$. From the high-precision photometry we obtain radius uncertaintiesof $\sim$1.6%, allowing us to conduct internal structure and atmospheric escapemodelling. TOI-1064 b is one of the densest, well-characterised sub-Neptunes,with a tenuous atmosphere that can be explained by the loss of a primordialenvelope following migration through the protoplanetary disc. It is likely thatTOI-1064 c has an extended atmosphere due to the tentative low density, howeverfurther RVs are needed to confirm this scenario and the similar radii,different masses nature of this system. The high-precision data and modellingof TOI-1064 b are important for planets in this region of mass-radius space,and it allows us to identify a trend in bulk density-stellar metallicity formassive sub-Neptunes that may hint at the formation of this population ofplanets.

A pair of Sub-Neptunes transiting the bright K-dwarf TOI-1064 characterised with CHEOPS