Context. We present the results of a new Chandra study of the galaxy cluster Abell 2626. The radio emission of the cluster shows a complex system of four symmetric arcs without known correlations with the thermal X-ray emission. The mirror symmetry of the radio arcs toward the center and the presence of two optical cores in the central galaxy suggested that they may be created by pairs of precessing radio jets powered by dual active galactic nuclei (AGNs) inside the core dominant galaxy. However, previous observations failed to observe the second jetted AGN and the spectral trend due to radiative age along the radio arcs, thus challenging this interpretation. Aim. The new Chandra observation had several scientific objectives, including the search for the second AGN that would support the jet precession model. We focus here on the detailed study of the local properties of the thermal and non-thermal emission in the proximity of the radio arcs, in order to obtain further insights into their origin. Methods. We performed a standard data reduction of the Chandra dataset deriving the radial profiles of temperature, density, pressure and cooling time of the intra-cluster medium. We further analyzed the two-dimensional (2D) distribution of the gas temperature, discovering that the south-western junction of the radio arcs surrounds the cool core of the cluster. Results. We studied the X-ray surface brightness and spectral profiles across the junction, finding a cold front spatially coincident with the radio arcs. This may suggest a connection between the sloshing of the thermal gas and the nature of the radio filaments, raising new scenarios for their origin. A tantalizing possibility is that the radio arcs trace the projection of a complex surface connecting the sites where electrons are most efficiently reaccelerated by the turbulence that is generated by the gas sloshing. In this case, diffuse emission embedded by the arcs and with extremely steep spectrum should be most visible at very low radio frequencies.