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X-ray surveys of UV-emitting AGB stars show that ∼40% of objects with FUV emission and GALEX FUV/NUV flux ratio <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>R</mi><mrow><mi>f</mi><mi>u</mi><mi>v</mi><mo>/</mo><mi>n</mi><mi>u</mi><mi>v</mi></mrow></msub></semantics></math></inline-formula> <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover><mstyle scriptlevel="1" displaystyle="false"><mo>∼</mo></mstyle><mo>></mo></mover></mrow></semantics></math></inline-formula> 0.2 (fuvAGB stars) have variable X-ray emission characterized by very high temperatures (Tx∼35–160 MK) and luminosities (Lx∼0.002–0.2 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mo>⊙</mo></msub></semantics></math></inline-formula>), indicating the presence of accretion associated with a close binary companion. However, the UV-emitting AGB star population is dominated by objects with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>R</mi><mrow><mi>f</mi><mi>u</mi><mi>v</mi><mo>/</mo><mi>n</mi><mi>u</mi><mi>v</mi></mrow></msub></semantics></math></inline-formula> ≲ 0.06 (nuvAGB stars), and we do not know whether the UV emission from these is intrinsic to the AGB star or extrinsic (i.e., due to binarity). In order to help distinguish between intrinsic and extrinsic models of the puzzling high-energy emission of cool AGB stars, we report results from two studies—(i) XMM-Newton X-observations of two nuvAGB stars, and (ii) simple chromosphere modeling. In study (i), we detect the one which has the lower FUV/NUV ratio, with a total Lx = 0.00027 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mo>⊙</mo></msub></semantics></math></inline-formula>, and a spectrum best fitted with a dominant component at Tx∼10 MK, most likely coronal emission from a main-sequence companion. Therefore, a significant fraction of nuvAGB stars may also be binaries with active, but weak accretion. Study (ii) shows that chromospheres with temperatures of ∼10,000 K can produce <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>R</mi><mrow><mi>f</mi><mi>u</mi><mi>v</mi><mo>/</mo><mi>n</mi><mi>u</mi><mi>v</mi></mrow></msub><mo>≲</mo><mn>0.06</mn></mrow></semantics></math></inline-formula>; higher ratios require hotter gas, implying active accretion.